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Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Confidence intervals Sayan Mukherjee Sta. 113 Chapter 7 of Devore November 8, 2007 Sayan Mukherjee Confidence intervals
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Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Table of contents 1 Normal distribution known variance 2 Large sample CI, or CLT to the rescue 3 Small sample normal, thank Guinness 4 Confidence intervals on the spread or variance 5 Confidence bounds 6 Sample size computations Sayan Mukherjee Confidence intervals
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Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Uncertainty In the last lecture we learned about point estimates using the MLE. We also learned about uncertainty in the context of Bayesian methods and the posterior density. We now study within the likelihood framework how to think of uncertainty. This is the idea of a confidence interval and in statistics lingo it is the frequentist analog of the Bayesian credible interval. Sayan Mukherjee Confidence intervals
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Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Confidence interval of the mean If X 1 , ..., X n iid No( μ, σ 2 ) with then we know that Z = ¯ X - μ σ/ n No(0 , 1) . This means that Pr ( - 1 . 96 < Z < 1 . 96) = . 95 . Pr - 1 . 96 < ¯ X - μ σ/ n < 1 . 96 ! = . 95 . Pr - 1 . 96 σ n < ¯ X - μ < 1 . 96 σ n ! = . 95 . Pr - 1 . 96 σ n - ¯ X < μ < - ¯ X + 1 . 96 σ n ! = . 95 . Pr 1 . 96 σ n + ¯ X > μ > ¯ X - 1 . 96 σ n ! = . 95 . Pr ¯ X - 1 . 96 σ n < μ < ¯ X + 1 . 96 σ n ! = . 95 . Sayan Mukherjee Confidence intervals
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Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations A random interval Consider the quantity Pr ¯ X - 1 . 96 σ n < μ < ¯ X + 1 . 96 σ n ! = . 95 , ¯ X is random but μ is not it is fixed. The interpretation of the above equation is as a random interval = ¯ X - 1 . 96 σ n , u = ¯ X + 1 . 96 σ n ! . The interval is centered at the sample mean and extends in either direction by 1 . 96 σ n . What a statistician would say is “ the probability is . 95 that the random interval includes the true value μ. Sayan Mukherjee Confidence intervals
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Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Formal definition Definition Given x 1 , ..., x n iid No ( μ, σ 2 ) compute ¯ x. The 95% confidence interval for μ is ¯ x - 1 . 96 σ n , ¯ x + 1 . 96 σ n , or as ¯ x 1 . 96 σ n .
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